Asymptotic models for scattering problems from unbounded media with high conductivity

We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no longer possible. Our new analysis is based on the use of Rellich-type estimates and boundedness of $L2$ solution operators. We also discuss numerical approximation of obtained GIBC (up to order 3) and numerically test theoretical convergence rates

Electromagnetic Wave Scattering from Rough Penetrable Layers

International audienceWe consider scattering of time-harmonic electromagnetic waves from an unbounded penetrable dielectric layer mounted on a perfectly conducting infinite plate. This model describes for instance propagation of monochromatic light through dielectric photonic assemblies mounted on a metal plate. We give a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space of functions defined in an unbounded domain containing the dielectric structure. Further, we derive a Rellich identity for a solution to the variational formulation. For simple material configurations and under suitable non-trapping and smoothness conditions, this integral identity allows to prove an a-priori estimate for such a solution. A-priori estimates for solutions to more complicated material configurations are then shown using a perturbation approach. While the estimates derived from the Rellich identity show that the electromagnetic rough surface scattering problem has at most one solution, a limiting absorption argument finally implies existence of a solution to the problem

DIRECT AND INVERSE MEDIUM SCATTERING IN A 3D HOMOGENEOUS PLANAR WAVEGUIDE

Time-harmonic acoustic waves in an ocean of finite height are modeled by the Helmholtz equation inside a layer with suitable boundary conditions. Scattering in this geometry features phenomena unknown in free space: resonances might occur at special frequencies and wave fields consist of partly evanescent modes. Inverse scattering in waveguides hence needs to cope with energy loss and limited aperture data due to the planar geometry. In this paper, we analyze direct wave scattering in a 3D planar waveguide and show that resonance frequencies do not exist for a certain class of bounded penetrable scatterers. More important, we propose the Factorization method for solving inverse scattering problems in the 3D waveguide. This fast inversion method requires near-field data for special incident fields and we rigorously show how to generate this data from standard point sources. Finally, we discuss our theoretical results in the light of numerical examples

Spectral Volumetric Integral Equation Methods for Acoustic Medium Scattering in a Planar Homogeneous 3D Waveguide

Scattering of acoustic waves from an inhomogeneous medium can be described by the Lippmann-Schwinger integral equation. For scattering problems in free space, Vainikko proposed a fast spectral solution method that exploits the convolution structure of this equation's integral operator by using the fast Fourier transform. In a planar 3--dimensional waveguide, the integral operator of the Lippmann-Schwinger integral equation fails to be a convolution. In this paper, we show that the separable structure of the kernel nevertheless allows to construct fast spectral collocation methods similar to Vainikko's technique. The numerical analysis of this method requires smooth material parameters; if the material parameters are, say, discontinuous, no theoretical statement on convergence is available. We show how to construct a Galerkin variant of Vainikko's method for which a rigorous convergence analysis is available even for discontinuous materials. For several distant scattering objects inside the 3--dimensional waveguide this discretization technique leads to a computational domain consisting of one large box containing all scatterers, and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects

An improved time domain linear sampling method for Robin and Neumann obstacles

We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We analyze this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation.L'objectif du travail présenté ici est l'analyse et la mise en place numérique d'une méthode d'échantillonnage linéaire (linear sampling method), permettant de résoudre le problème de diffraction inverse suivant : à partir de mesures dans le domaine temporel d'ondes diffractées causales, retrouver la forme et la position de l'obstacle diffractant. Nous avons pour objectif d'étendre le travail effectué un travail antérieur au cas où l'on soumet le bord de l'obstacle à une condition de type Robin-Fourier. Le cadre de travail est également étendu au cas où les ondes incidentes ont un spectre significatif borné. Sur le plan numérique, nous proposons une méthode permettant de reconstruire avec la même précision des obstacles avec des conditions mixtes, de type Dirichlet, Neumann ou Robin-Fourier

A non-iterative sampling approach using noise subspace projection for EIT

International audienceThis study concerns the problem of the reconstruction of inclusions embedded in a conductive medium in the context of electrical impedance tomography (EIT), which is investigated within the framework of a non-iterative sampling approach. This type of identification strategy relies on the construction of a special indicator function that takes, roughly speaking, small values outside the inclusion and large values inside. Such a function is constructed in this paper from the projection of a fundamental singular solution onto the space spanned by the singular vectors associated with some of the smallest singular values of the data-to-measurement operator. The behavior of the novel indicator function is analyzed. For a subsequent implementation in a discrete setting, the quality of classical finite-dimensional approximations of the measurement operator is discussed. The robustness of this approach is also analyzed when only noisy spectral information is available. Finally, this identification method is implemented numerically and experimentally, and its efficiency is discussed on a set of, partly experimental, examples